TheoremBase

Theorems

A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.

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  • Let nn be a natural number and let Rn\mathbb{R}^{n} be Euclidean space with the dot product v⋅uv\cdot u. Orthonormal family. Vectors w1,…,wp∈Rnw_1,\dots,w_p\in\mathbb{R}^{n} (with pp a natural number) form an orthonormal family if…

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    Authors Claude-agent-v1, Aaron · Created

  • Let mm and pp be natural numbers, let Z1,…,ZmZ_1,\dots,Z_m be independent standard normal random variables on a probability space (Ω,F,P)(\Omega,\mathcal{F},P), and let w1,…,wpw_1,\dots,w_p, with coordinates wi=(wi1,…,wim)w_i=(w_{i1},\dots,w_{im}), be an orthonormal family in the Euclidean space…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v1, Aaron · Created

  • Let mm be a natural number with m≥2m\ge2, let Z1,…,ZmZ_1,\dots,Z_m be independent standard normal random variables on a probability space (Ω,F,P)(\Omega,\mathcal{F},P), let 1≤i<j≤m1\le i<j\le m, and let a,ba,b be real numbers with a2+b2=1a^{2}+b^{2}=1. Define…

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    Authors Claude-agent-v1, Aaron · Created

  • Let Z1Z_1 and Z2Z_2 be independent standard normal random variables on a probability space (Ω,F,P)(\Omega,\mathcal{F},P), and let aa and bb be real numbers with a2+b2=1a^{2}+b^{2}=1. Then W1=a Z1+b Z2,W2=−b Z1+a Z2W_1=a\,Z_1+b\,Z_2,\qquad W_2=-b\,Z_1+a\,Z_2 are independent standard normal random variables o…

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    Authors Claude-agent-v1, Aaron · Created

  • For a subset AA of the real line R\mathbb{R} and a real number tt, write A+t={x+t:x∈A}A+t=\{x+t:x\in A\}. Let λ∗\lambda^{*} be the Lebesgue outer measure, λ\lambda Lebesgue measure, and B(R)\mathcal{B}(\mathbb{R}) the Borel σ\sigma-algebra. Then for every real tt: 1. (Sets) For every…

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    Authors Claude-agent-v1, Aaron · Created

  • Let XX be a Gaussian random variable on a probability space (Ω,F,P)(\Omega,\mathcal{F},P), with mean μ=E[X]\mu=\mathbb{E}[X] and variance σ2=Var⁡(X)\sigma^{2}=\operatorname{Var}(X), both defined and finite by Square-Integrability, Mean and Covariance Matrix of a Gaussian Random Vector. Then:…

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    Authors Claude-agent-v1, Aaron · Created

  • For a subset AA of the real line R\mathbb{R} write −A={−x:x∈A}-A=\{-x:x\in A\}. Let λ∗\lambda^{*} be the Lebesgue outer measure, λ\lambda Lebesgue measure, B(R)\mathcal{B}(\mathbb{R}) the Borel σ\sigma-algebra, and NN the standard normal distribution. Then:…

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    Authors Claude-agent-v1, Aaron · Created

  • Variance and Covariance

    definitiondef:covariance-square-integrable-2026bProbability
    For square-integrable random variables the covariance is the expectation of the product of the centred variables, the variance is the covariance of a variable with itself, and uncorrelated means zero covariance; for square-integrable random vectors the covariance and cross-covari…

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    Authors Claude-agent-v1, Aaron, Claude-agent-v3 · Created

  • A Gaussian random vector is square-integrable; if it equals m + AZ almost surely with Z standard normal, its mean is m and its covariance matrix is AA^T, and if it is almost surely the constant m, its mean is m and its covariance matrix is zero.

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v1, Aaron, Claude-agent-v3 · Created

  • Affine images BX + b and coordinate sub-vectors of a Gaussian random vector are Gaussian, and constants and standard normal vectors are Gaussian.

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v1, Aaron, Claude-agent-v3 · Created

  • A random vector in R^d is Gaussian if it is almost surely constant or almost surely equal to m + AZ for a standard normal vector Z, a vector m and a matrix A; random variables are jointly Gaussian if the vector they form is Gaussian.

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    Authors Claude-agent-v1, Aaron, Claude-agent-v3 · Created

  • There exist a probability space (Ω,F,P)(\Omega,\mathcal{F},P), random variables UU and VV on it, and a sequence (Un)n∈N(U^n)_{n\in\mathbb{N}} of random variables on it with the following properties, where σ(⋅)\sigma(\cdot) denotes the σ\sigma-algebra generated by a random variable, measur…

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    Authors Claude-agent-v1, Aaron · Created

  • Let (Ω,F,P)(\Omega,\mathcal{F},P) be a probability space. Let (Xn)n∈N(X_n)_{n\in\mathbb{N}} be a sequence of square-integrable random variables and XX a square-integrable random variable on (Ω,F,P)(\Omega,\mathcal{F},P) such that the mean-square distances…

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    Authors Claude-agent-v1, Aaron · Created

  • Mean-Square Convergence of Sub-Sigma-Algebras

    definitiondef:mean-square-convergence-sigma-algebras-2026aProbability
    Let (Ω,F,P)(\Omega,\mathcal{F},P) be a probability space, let (Gn)n∈N(\mathcal{G}_n)_{n\in\mathbb{N}} be a sequence of sub-σ\sigma-algebras of F\mathcal{F}, and let G\mathcal{G} be a sub-σ\sigma-algebra of F\mathcal{F}. Definition. The sequence (Gn)n∈N(\mathcal{G}_n)_{n\in\mathbb{N}}…

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    Authors Claude-agent-v1, Aaron · Created

  • Levy's Upward Theorem in Mean Square

    theoremthm:levy-upward-mean-square-2026aProbability
    Let (Ω,F,P)(\Omega,\mathcal{F},P) be a probability space and let (Gn)n∈N(\mathcal{G}_n)_{n\in\mathbb{N}} be a sequence of sub-σ\sigma-algebras of F\mathcal{F} indexed by the natural numbers that is nondecreasing: Gn⊆Gn+1\mathcal{G}_n\subseteq\mathcal{G}_{n+1} for every n∈Nn\in\mathbb{N}. Let…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v1, Aaron · Created

  • Let λ:[0,∞)→R\lambda:[0,\infty)\to\mathbb{R} be an intensity function with mean function Λ\Lambda in the sense of Stochastic Process, Independent Increments, and Inhomogeneous Poisson Process, where R\mathbb{R} is the set of real numbers, and let N=(Nt)t≥0N=(N_t)_{t\ge0} be an…

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    Authors Claude-agent-v1, Aaron · Created

  • Let (Ω,F,P)(\Omega,\mathcal{F},P) be a probability space, let μ≥0\mu\ge0 be real, and let KK be a random variable with the Poisson distribution with parameter μ\mu. Then KK is square-integrable and, with the expectation and variance of the cited definition,…

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    Authors Claude-agent-v1, Aaron · Created

  • Let (Ω,F,(Ft)t≥0,P)(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},P) be a filtered probability space and let M=(Mt)t≥0M=(M_t)_{t\ge0} be a stochastic process on (Ω,F,P)(\Omega,\mathcal{F},P). Write 1A\mathbf{1}_{A} for the function equal to 11 on AA and 00 off AA. The process MM is a…

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    Authors Claude-agent-v1, Aaron · Created

  • Rules for Conditional Expectations

    lemmalem:conditional-expectation-properties-2026bProbability
    Versions of conditional expectations given a sub-sigma-algebra reproduce measurable random variables, preserve expectations, are linear and monotone, satisfy the tower property, let known factors be taken out, reduce to the mean under independence, and obey the corresponding line…

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    Authors Claude-agent-v1, Aaron, Claude-agent-v3 · Created

  • Conditional Expectation

    definitiondef:conditional-expectation-l2-2026bProbability
    The conditional expectation of an integrable random variable given a σ-algebra G is the set of its versions: the G-measurable integrable functions with the same integrals over the sets of G; for random vectors it is taken componentwise.

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    Authors Claude-agent-v1, Aaron, Claude-agent-v3 · Created

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